It has ever been a claim of the Catholic Church that it is the most
democratic society that the world has yet seen.
– Thomas Adolphus Trollope, "The papal conclaves," 1876, page 5.

We admit that the Papacy falls short of what most people would consider a "democracy"
(e.g. only Cardinals vote to elect the Pope, and both they and the Pope are all-male).
But there is no question
that among all human positions of high power that are elected,
the Papacy is the one that has outlasted every other on the face of the earth.

Approval Voting was used for
selecting 41 Popes, in a system instituted by Pope (Saint)
Celestine V in 1294, and which remained in force until 1621.

How this system came about

Of the extent to which the [popedom] was becoming more and more a political institution, we
have the striking evidence in the brief pontificate of CELESTINE V (1294).
A hermit of the Abruzzi, of austere and holy life,
he had been elected pope in the hope that his reputation for virtue might in some measure
restore the character of his office. Something more, however, than mere
sanctity and blamelessness were now necessary for the discharge of the duties of a position
which
by its associations demanded the exercise of statecraft, political intrigue,
and a wide knowledge of affairs. In less than six months Celestine resigned an office for
which by lack of any experience and ability he was altogether unfitted...
–
1897 Encylopedia Brittanica, "Popedom."

The way this reform was enacted was interesting.
The conclave trying to determine the new pope had been hopelessly deadlocked
for over two years, unable to find anybody they were willing to grant
the necessary two-thirds supermajority plurality vote.
Finally, in desperation,
they came up with
the obscure Benedictine monk
Pietro di Murrone (sometimes called
Pietro Angelerio, and there are several variant spellings of both names).
This 81-to-85-year-old hermit had literally been living in a cave on Mount Maiella.
He fasted every day except sunday.
He'd achieved a certain fame as a role model for a sect of
Christian ascetic fanatics called the "Celestines," who had
formed, in all, about 600 monasteries.
The joy of appointing him was that he could not
be identified with any of the big rival factions inside the church and was the most incorruptible
person they could think of. Therefore, the conclavist factions could
reassure themselves that he probably wouldn't associate himself with the other side.
So in July 1294, three eminent dignitaries accompanied by a vast crowd of monks and laymen,
ascended Mount Maiella to announce that Pietro had unanimously been chosen pope!
He was humbly begged to cease his hermit-ways to be crowned pope Celestine V.
(Note: after somebody is crowned pope, they take a different name.)

Now there was a reason Pietro was a hermit. He liked solitude. He was inclined
to stay in his cave. However, after reflection and prayer,
he heard the Voice of God commanding him to sacrifice his personal
desires for the welfare of the church (which clearly was in trouble), and accepted.

Since it was obvious that the election process for popes was dysfunctional (the conclave in
1271 that chose Gregory X had been deadlocked even longer than in 1294!),
Celestine invented, and enacted by decree, the approval-voting based system.
William Poundstone in Gaming the Vote speculates
his rules probably were inspired by the Venetian
system and cites a paper by
J.M. Colomer & Iain McLean:
Electing popes, approval balloting and qualified-majority rule (pdf),
J.Interdisciplinary History 29,1 (1998) 1-22; reprinted in
Politics and Political Change (MIT Press 2001, R.I.Rotberg ed.) 47-68.
They in turn cite the "Ordinarium Sanctae Romanae Eclesiae" by Cardinal Jacobi Gaytani.

As regards Celestine's other moves as pope:
Unfortunately for him, Celestine was naive and unaware of, and/or ignored,
the vast rivalries dividing the church.
They required careful handling.
Popes of that day were generally smooth, complex, corrupt manipulators substantially concerned
with enriching themselves.
Celestine was not.
He lavishly handed out favors and appointments, especially to Neopolitans (horrors!)
and was generous with the Church's
resources, while annoyingly not consulting powerful cardinals;
he couldn't handle heavy financial and political complexities by prayer and fasting;
and corruption supposedly ran riot under his lax supervision.

Both Celestine and the Cardinals were uncomfortable with, and perceived the inadequacy of,
his exercise of power.

So after instituting his election reform, Celestine determined to
become one of the few (in fact, I would contend, the only) Popes ever to resign.
However, while Celestine's moves had made him unpopular with some of the church's most
powerful officials, he was immensely popular with the general public –
as well as with
King Charles II of Naples, who'd benefited greatly from Celestine – e.g.
Celestine had appointed Charles's 21-year-old son Louis the archbishop of Lyon.

Vast crowds assembled imploring Celestine to remain in office.
Nevertheless, he insisted on announcing his resignation on 13 December 1294.
Celestine then intended to return to the monastery and his humble hermit life.
However, after an unsuccessful escape attempt, he was
arrested and imprisoned by his successor Boniface VIII, for fear Celestine V might
reclaim the papacy!
More likely, Boniface feared Celestine might be reinstalled by others somehow, or that his
supporters might schism the church, or that Celestine together with them might do something
unpredictable.
Or: Boniface, once he had his hand on the reins of power, did not want to take the slightest chance
they would be wrested from him.
(Boniface, incidentally, was the complete opposite of Celestine; he was trained as a lawyer,
had a long career as a diplomat, and carried on lifelong feuds.)
Celestine died in prison on 19 May 1296 – and there
has been speculation Boniface had him assassinated.
Indeed, Celestine's skull contains a suspicious nail-sized hole, see photo at left.

Boniface then revoked almost all Celestine's papal decisions and appointments
(since they had been made in "the fullness of his simplicity") –
but Celestine's approval-voting-based election rules were not revoked.
They remained in force for 329 years until revoked in 1621 by Gregory XV.

The power of the Pope

The primacy [of the papacy] over the entire church was conferred by Christ on Peter;
and this primacy was one, not of mere dignity, but of full and supreme jurisdiction.
–
1964 Encylopedia Americana, "Papacy."

But while, with respect to the acceptance of doctrine, the losses of the
16th century were thus materially retrieved,
the popedom was sinking rapidly in political importance.
Its influence in [Italy] dwindled to within the limits of the [Papal states];
the dynastic successions in Naples and Sicily underwent a total change without the
curia.. being in any way consulted... and Clement XL's chagrin, when he found himself
compelled to recognize the
pretensions of the archduke Charles to the Spanish throne, was intense.

...The conviction had long been growing in the chief cities of the
Continent that whenever the representatives of Jesuitism obtained a footing the cause of
public order and domestic peace was placed in jeopardy...
Toleration was extended to protestant sects and the Greek church...
–
1897 Encylopedia Brittanica, "Popedom."

In the period we are considering, the Pope was very powerful.
And not just indirectly via thinking holy thoughts;
directly, via taxing the populace, ruling, owning vast wealth,
having armies and declaring war.
E.g. Pope Julius II took to the field in full armor and
conquered most of Romagna, captured Bologna and Perugia,
and successfully battled Venice, forcing it to cede Ravenna, Cervia, Faenza, and Rimini.
(The popes also, of course, played a major role in the "crusades.")

Between
756 and 1870
the pope ruled over his own country, known as the "papal states,"
consisting of
chunks of what is now Italy and France.
Popes also crowned emperors (e.g. Charlemagne), thus conferring "legitimacy" upon them,
"authorized" others to start wars (causing said wars to have a much greater chance of
success), adjudicated disputes between them, and loaned others
vast sums of money enabling them to
continue wars. The popes also controlled the marriages and divorces of kings,
often in return for favors.
Between all that, the vast worldwide
revenues of the church, its spiritual influence, and the
pope's essentially dictatorial power over the whole church
– the pope at peak,
on balance, was one of the most powerful, perhaps even the
most powerful, person in the western world.

And as an example of the popes' power just via moral authority alone,
Nicholas V and Callixtus III in the mid 1400s both
authorized Portugal to enslave Africans and generally legitimized the slave trade
and the practice of hereditary slavery. If they
instead had come down against slavery,
world history might have altered.

Even nowadays, while the
power of the papacy has shrunk greatly relative to secular governments,
it remains considerable.
The CIA estimates there are 1 billion Catholics, i.e. about 16% of the world population.
(Also about 24% of the USA's population professes Catholicism.)
The annual revenues of the Catholic Church
are hard for me to estimate – they do not
publish budgets and shroud a lot of their finances in secrecy.
[E.g. when the Massachussetts government attempted to make a law saying all churches
with annual revenues exceeding
$500,000 had to disclose their finances, the bill was vigorously resisted by
the Catholic Church, see Scott Helman and Frank Phillips:
O'MALLEY RESISTS BILL ON FINANCIAL DISCLOSURE,
Boston Globe 21 Jan. 2006 page B1.]
One estimate (as of 2008) is that they are
in the neighborhood of $500 billion.
The pope, in principle, has complete control over that money.
This figure far exceeds the GNPs of many countries
(e.g. Ethiopia $75 billion, Burma/Myanmar $100 billion, Greece $256 billion, Peru $190 billion;
the United Nations budget is below $2 billion).
On the other hand, I also can make revenue estimates as low as $20 billion!

A Reuters article
Deborah Zabarenko: "After scandal, fiscal troubles deepen for US Catholic Church,"
published in the Boston Globe 29 December 2004 said
"...the US Catholic Church's annual revenue...in 2001 was an estimated $102 billion."
This would extrapolate (based on USA versus world GDP) to about $500 billion
worldwide revenues. On the other hand, the
Archdiocese of Los Angeles (the largest in the USA) reported (Fiscal Year ending 30 June 2007)
$147,664,560 in revenues (considerably exceeded by expenses)
which extrapolates to only $4.4 billion USA-wide revenues and about $22 billion worldwide.
Approximately 1 percent of world population, at any given time, is enrolled as students
in Catholic schools, with typical tuition of $5000-$7000 per student in the USA
2007-8 school year.
This revenue source alone (extrapolated from the USA's 2.3 million students in 2006-7
to worldwide via GDP ratio) adds up
to about $70 billion per year. I conclude from this that reports of "diocese revenues" have
little to do with actual revenues. (Catholic schools are run by dioceses.)
The Pope's once-huge dominion is nowadays reduced to little more than
Vatican city with population 824 (July 2008 est.) and land area about 1 square kilometer.
It was claimed by the CIA world factbook to have annual revenues of $310 million (in 2006)
versus $307 million in expenditures. This however again is
not very indicative of the worldwide
picture.

We should note that some of the
Popes during the period we are discussing
were not (as they are reputed to be today) celibate and sinless.
Some had quite a few offspring (e.g. Pius IV with three illegitimate children,
Innocent VIII with a son and daughter who lived with him in the Vatican,
Alexander VI with 7-10 illegitimate children)
and/or were into big-money nepotism, bribery, torture, extortion,
and murder.

Also, a number of "antipopes" were elected
by factions, then claimed to be the pope and regarded the "true" pope
as illegitimate.
Antipope Anacletus II was elected by a majority
while his rival Innocent II (served 1130-1143) was elected by a
minority, but the latter nevertheless eventually came to be regarded as the
"true pope." That all was before the period we are discussing, however.
The "great schism" came from
the allegedly invalid election of Pope Urban VI.
After his election, Urban exhibited great arrogance and immoderation.
Many historians believe he was insane. The cardinals who elected him declared that
his was not a genuine election and had been the result of their fear of the surrounding
Roman populace (who had demanded the return of the popedom to Rome).
The Cardinals declared their earlier election of Urban VI null and void
(eventually 15 of the 16 who'd voted for Urban renounced him, and the 16th died)
and
instead in 1378 elected the antipope Clement VII.
Urban VI refused to accept that.
Clement took up residence in Avignon France while Urban continued to rule from Rome,
and they each excommunicated the other.
This led to two, and eventually three, rival lines of claimants to papacy:
the Roman, Avignon, and Pisan line. The schism was re-unified and healed 39 years later, however.

How Celestine's election rules worked

Elections typically had about
40 voters (Cardinals) and 20 candidates.
Secret-ballot approval votingbut with a 2/3 supermajority requirement
was employed, with re-voting as many times as necessary (while
also introducing new candidates into the race as necessary) until
the winner exceeded that threshold.

Unlike in many elections elsewhere, it has often happened that the Pope was
a member of the conclave that elected him.
That enabled him to vote for himself,
and also to engage in deals and negotations, possibly interacting with other Pope-candidates
doing the same. (However, there were also other elections in which the winner was not present and
indeed whose election came as a complete surprise to him.)
A note of minor but nonzero importance:
The threshold of 2/3 was one vote larger in situations in which a pope could
vote for himself.

Some Pope elections were heavily influenced by bribery,
called in the ecclesiastical context "simony."

Unfortunately for our purposes, it
can be hard to tell what went on at the elections because the
conclavists were (and still are) sworn to secrecy, met in a private location,
and famously burned all the ballots
at the end (to create the "smoke signal" announcing the new Pope).

However, during this period (especially after 1400) conclave secrecy was generally
quite imperfect and in particular "ambassadors" from outside would
write reports back to their masters describing what was going on in
the conclaves. Some conclavist-cardinals were pretty much tools of various Kings.
From such letters and reports, quite a lot is known about what happened at some
conclaves – especially the interesting and controversial ones.

Electoral speed
was encouraged by the "Ubi Periculum" rules (originally due to Gregory X although
almost immediately cancelled by his successor but renewed by Celestine V)
as these: if it took longer than 3 days, the cardinals
were restricted to 2 simple meals per day; if it took longer than 8, their diet was
reduced to bread and water only; and the conclave was usually held in austere quarters –
a single large room with cots for sleeping,
plus a single lavatory, and that was all, with
armed guards.
One of the new
rules after the approval-voting-based system was abandoned
was that the kings of France and Spain – and later Austria – were granted
veto power over the election of
any pope they found displeasing, albeit they only got to exercise one veto per election.
This "jus exclusiva" was in force until about 1910.

During
repeated re-votes, the totals for different candidates could and sometimes did change
a lot as new information arose and/or as the strategic environment changed.
(For example, one Pope initially got zero approvals; and many times,
pope-candidates who got over 50% approval failed to win – since of course
the ultimate winner got still more approval.)
One final note: sometimes the number of voters changed
during the conclave because,
e.g. some conclavists became sick or died
during the proceedings; or because of late-arriving
cardinals.

The system used right now (2008) to elect Popes

The approval-voting system is no longer used, nor any longer
are the quarters for the conclavists austere,
nor are their victuals anymore rationed nor plain.
The rules
being used right now instead
involve plain plurality-style voting
with a two-thirds supermajority requirement.
The voters are cardinals below age 80
(typically about 100 to 120 in number). They conclave in the Sistine Chapel.

If after 13 days (and 26 ballots – two per day)
a pope has not been elected, the cardinals pause for a day of
prayer and reflection, and the two candidates with the most votes in the
latest balloting will be in a runoff election (but these two cannot vote).
The new pope must garner two-thirds of the vote in that runoff.

Alternatively, the cardinals can decide by simple majority vote to choose
a pontiff by simple majority vote (but note a majority, not merely a plurality,
is required).

The present rules strike me as worse than Celestine's 1294 rules because they are susceptible
to "vote-splitting," "wasted vote," "spoiler," "nonmonotonicity,"
and "momentum" pathologies.
They may, however, have the advantage of being less
susceptible to permanent deadlock.
The Catholic Church is not noted for taking my advice, but it would be to
replace plurality with approval or range voting.

Sources

Pope Leo XIII opened the Vatican Archives to scholars around the world
(whether Catholic or not) in about 1900.